Parallel Rank of Two Sandpile Models of Signed Integer Partitions

Abstract

We introduce the concept of fundamental sequence for a finite graded poset X which is also a discrete dynamical model. The concept of fundamental sequence is a refinement of the concept of parallel convergence time for these models. We compute the parallel convergence time and the fundamental sequence when X is the finite lattice $P(n,r)$ of all the signed integer partitions $a_r,\dots ,a_1,b_1,\dots ,b_{n-r}$ such that $r\ge a_r\ge \cdots \ge a_1\ge 0\ge b_1\ge \cdots \ge b_{n-r}\ge -(n-r)$, where $n\ge r\ge 0$, and when X is the sublattice $P(n,d,r)$ of all the signed integer partitions of $P(n,r)$ having exactly d nonzero parts.